flow control kaist cs644 advanced topics in networking
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Flow Control KAIST CS644
Advanced Topics in Networking
Jeonghoon Mo<jhmo@icu.ac.kr>
School of EngineeringInformation and Communications University
2Jeonghoon MoOctober 2004
Acknowledgements
Part of slides is from tutorial of R. Gibbens and P. Key at SIGCOM
M 2000 S. Low’s OFC presentation
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Overview
Problem Objectives Kelly’s Framework - Wired Data Networks Extensions Quality of Service Wireless Network High Speed Network: Aggregated Flow
Control
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Problem
How to control the network to share the bandwidth efficiently and fairly?
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Link Model
Set of resources, J; set of routes, R A route r is a subset r J. Let
Capacity of resource j is Cj.
otherwise0
1 rjA jr
A’x c x 0
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A Few System Objectives
Max Throughput Max-min Fairness (Most Common) Proportional Fairness (Kelly) -Fairness (Mo, Walrand)
8Jeonghoon MoOctober 2004
Max System Throughput
Maximize: x(1) + x(2) + x(3)
x*= (0,6,6) maximizes the total system throughput.
However, user 1 does not get anything. => unfair
6 6x1x2 x3
Two links with capacity 6 Three users: 1,2,3 x(i) : bandwidth to user i x(1)+x(2) <= 6 x(1)+x(3) <= 6
9Jeonghoon MoOctober 2004
Max-Min Fairness
Most commonly used definition of fairness. Maximize Minimum of the x(i), recursively. x*= (3,3,3) is the max-min allocation.
However, user 1 uses more resources.
6 6x1x2 x3
10Jeonghoon MoOctober 2004
Proportional Fairness
Proposed by Frank Kelly Social Welfare: Sum of Utilities of Users Maximize the Social Welfare x*= (2,4,4) is the Proportional Fair Allocation. Can be generalized into “Utility Fairness”.
6 6x1x2 x3
i
ix )log(
11Jeonghoon MoOctober 2004
-Fairness
Generalized Fairness Definition System Objective:
includes proportional-fair, max-min-fair, max throughput = 0 : Maximum allocation (p=1) 1 : Proportional fair allocation = 2 : TCP-fair allocation : Max-min fair allocation
(p=1)
Max i pix(1-)
1-
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-Fairness
Trade-off between Fairness and Efficiency Bigger favors Fairness Smaller favors Efficiency
(source: Is Fair Allocation Inefficient, INFOCOM 04)
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Players
source controls its rate or window based on
(implicit or explicit) network feedback router (link) Generate (implicit) feedback or
controls packets
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User: rate and utility
Each route has a user: if xr is the rate on route r, then the utility to user r is Ur(xr).
Ur() --- increasing, strictly concave, continuously differentiable on xr [0 , ) --- elastic traffic
Let C=(Cj, j J), x=(xr, r R) then Ax C.
20Jeonghoon MoOctober 2004
System problem
Maximize aggregate utility, subject to capacity constraints
0over
subject to
max
x
CAx
xURr
rr
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User problem
User r chooses an amount to pay per unit time wr, and receives in return a flow xr = wr/r
0over
max
w
wwU
r
r
r
rr
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Network problem
As if the network maximizes a logarithmic utility function, but with constants (wr, rR) chosen by the users
0over
subject to
logmax
x
CAx
xwRr
rr
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Decomposition theorem
There exist vectors , w and x such that
1. wr = rxr for r R
2. wr solves USERr(Ur; r)
3. x solves NETWORK(A, C; w)
The vector x then also solves SYSTEM(U, A, C).
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Thus the system problem may be solved by solving simultaneously the network and user problems
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Result
A vector x solves NETWORK(A, C; w) if and only if it is proportionally fair per unit charge
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Solution of network problem
Strategy: design algorithms to implement proportional fairness
Several algorithms possible: try to mimic design choices made in existing standards
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Interpretation of primal algorithm
Resource j generates feedback signals at rate j(t)
signals sent to each user r whose route passes through resource j
multiplicative decrease in flow xr at rate proportional to stream of feedback signals received
linear increase in flow xr at rate proportional to wr
30Jeonghoon MoOctober 2004
Related Work
Optimization Flow Control (S. Low) Window based Model (Mo, Walrand)
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Optimization Flow Control
Distributed algorithm to share network resources
Link algorithm: what to feed back RED
Source algorithm: how to react TCP Tahoe, TCP Reno, TCP Vegas
Source alg
Link alg
32Jeonghoon MoOctober 2004
Welfare maximization
Primal problem:
Capacity can be less than real link capacity
Primal problem hard to solve & does not adapt
Llcx
xU
lSsls
sss
Mxm sss
, subject to
)( max
)(
lc
33Jeonghoon MoOctober 2004
c1 c2
Model
Network: Links l each of capacity cl
Sources s: (L(s), Us(xs), ms, Ms)
L(s) - links used by source sUs(xs) - utility if source rate = xs
x1
x2
x3
sss Mxm
121 cxx 231 cxx
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Distributed Solution
Dual problem:
BW price along path of s
Given sources can max own benefit individually indeed primal optimal if is dual optimal Solve dual problem!
pD(p)
0min
s l
lls
s cppBpD )()(
p psl
l L s
( )
B p U x x pss
m x Ms s s
s
s s s
( ) ( )
max
ps
x pss( ) ps
35Jeonghoon MoOctober 2004
Dual problem:
Grad projection alg:
Update rule:
A distributed computation system to solve the dual problem by gradient projection algorithm
Distributed Solution (cont…)
D(p)p
min0
p t p t D p t( ) [ ( ) ( ( ))] 1
))(( )1(
)])(()([ )1(1' tpUtx
ctxtptps
ss
ll
ll
36Jeonghoon MoOctober 2004
Source Algorithm
Decentralized: Source s needs only and )(' ss xU ps
ps
x p U pss
ss( ) ( )'= - 1
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Router (Link) Algorithm
Decentralized Rule of supply and demand Any work-conserving service discipline Simple
)])(()([)1( ll
ll ctxtptp
aggregate source rate
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Random Exponential Marking (REM)
Source algorithm Identical but does not communicate source rate
Link algorithm At update time t, sets price to a fraction of buffer
occupancy:
Theorem: Synchronous convergence Under same conditions (with possibly smaller ) :
Price update maintains descent direction Gradient estimate converge to true gradient Limit point is primal-dual optimal
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RED
Idea: early warning of congestion Algorithm
Link: Source (Reno):
Bqueue
marking
1
time
window
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RED
Idea: marks for estimation of shadow price Algorithm
Link Source
Global behavior of network of REM: stochastic gradient algorithm to solve dual problem
Q
queue
marking
1
1
fraction of marks
rate
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Window-based Model [Mo,Walrand]
x1 + x2 c1q1(c1 - x1 - x2) =
0w1 = x1 d1 + x1 q1 + x1 q2
xi 0, i = 1, 2, 3qi 0, i = 1, 2,
A’x c
Q(c - A’x) = 0w = X(d + qA)
c1 c2x1
x2 x3
q11
q12q23
q21
d1
d2 d3
w1
Q = diag{qi }; X = diag{xi }.
42Jeonghoon MoOctober 2004
Window-based AlgorithmTheorem:[Mowlr98]
Then x(t) -> unique weighted -fair point x*
Proof:
The function (si /wi ) 2i is a Lyapunov function
Let dwi
dt = - k di si
ti witi := end-to-end delay
si := wi - xi di - pi
43Jeonghoon MoOctober 2004
Extensions
Aggregated Flow Control Quality of Service Wireless Network Maxnet and Sumnet
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Aggregate Flow Control
Motivations: High Capacity of Optical Fiber
Idea: player are core routers and access
routers.access router: regulates the rate of
aggregated flowcore router: provide feedbacks to access
routers
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Quality of Service
Only bandwidth is modeled. QoS is affected by loss and delay also
How to incorporate other parameters?
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Non-Convex Utility Function (Lee04)
Considered sigmoidal utility function
Non-convex optimization problem =>duality gap
(source: J. Lee et. al. Non-convexity Issues, INFOCOM 04)
47Jeonghoon MoOctober 2004
Non-Convex Utility Functions
(source: J. Lee et. al. Non-convexity Issues, INFOCOM 04)
Dual Algorithm withSelf-Regulating Property
With Self-Regulation
Without Self-Regulation
48Jeonghoon MoOctober 2004
Wireless Ad-Hoc Network [RAD04]
Physical Model:Rate r is an increasing function of SINR. MAC : Each time slot determines power pn,which determine
s rate xn
Routing matrix R and flow to path matrix F are given.
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Random Topology Results
100m x100m grid
12 random node, with 6 pairs of transmissions
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In the wireless Ad-hoc Networks
The max-min fair rate allocation of any network has all rates equal to the worst node.
The capacity maximization objective leads to starving users.
Proportional Fair Allocation give reasonable trade-off between fairness and efficiency. The worst node does not starve.
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